A zoo of Hopf algebras Mike Zabrocki York University Non-Commutative Symmetric Functions I: I will present an overview of what Combinatorial Hopf Algebras (CHAs) are about by introducing definitions and examples. I will try to show how examples of CHAs are related to each other and where they can appear in the literature before they are recognized as CHAs. I will also give some examples where these objects appear in other areas of mathematics.
Transcript
A zoo of Hopf algebras
Mike ZabrockiYork University
Non-Commutative Symmetric Functions I:
I will present an overview of what Combinatorial Hopf Algebras (CHAs) are about by introducing definitions and examples. I will try to show how examples of CHAs are related to each other and where they can appear in the literature before they are recognized as CHAs. I will also give some examples where these objects appear in other areas of mathematics.
Partitions
Unordered lists of non-negative integers
Partitions
Unordered lists of non-negative integers
Partitions
Unordered lists of non-negative integers
Partitions
Unordered lists of non-negative integers
Create an algebra
Create an algebra
linearly spanned by partitions
Create an algebra
linearly spanned by partitions
Commutative product
Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
linear span of partitions of size
Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
linear span of partitions of size
Define a commutative product
Define a commutative product
Define a commutative product
Define a commutative product
properties of this algebra
Commutative and graded
properties of this algebra
Commutative and graded
properties of this algebra
Commutative and graded
Properties of this algebraFreely (commutative) generated by building blocks of rows
Properties of this algebraFreely (commutative) generated by building blocks of rows
Properties of this algebraFreely (commutative) generated by building blocks of rows
properties of this algebra
properties of this algebra
Bilinear
properties of this algebra
Bilinear
properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
From combinatorial object to algebra
partitions
From combinatorial object to algebra
partitions
some commutative
product
From combinatorial object to algebra
partitions
some commutative
product
Algebra
From combinatorial object to algebra
partitions
Algebraanother commutative
product
A different commutative product
And yet the algebra which arises isisomorphic to
A different commutative product
And yet the algebra which arises isisomorphic to
This algebra is special
This algebra is special
Just about any algebra with basis indexed by partitions is Isomorphic: e.g. symmetric functions, representation ring of symmetric group, ring of characters of GLn modules, cohomology rings of the grassmannians, etc.
This algebra is special
Just about any algebra with basis indexed by partitions is Isomorphic: e.g. symmetric functions, representation ring of symmetric group, ring of characters of GLn modules, cohomology rings of the grassmannians, etc.
Has a Hopf algebra structureproduct + coproduct + antipodewhich all interact nicely with each other
What is a Hopf algebra?
What is a Hopf algebra?Start with a Bialgebra
What is a Hopf algebra?
Product
Coproduct
Start with a Bialgebra
What is a Hopf algebra?
Product
Coproduct
with unit
and counit
Start with a Bialgebra
What is a Hopf algebra?
Product
Coproduct
with unit
and an antipode map
and counit
Start with a Bialgebra
What is a Hopf algebra?
this diagram commutes
Start with a Bialgebra
What is so good about a Hopf algebra?
The graded kind associated with combinatorial objects have lots of structure
There seems to be just “one” graded combinatorial hopf algebra for each type of combinatorial object
Many of the combinatorial operations are reflected in the algebraic structure
What is so good about a Hopf algebra?
There is “usually” an internal product structure and on some bases of some algebras this is hard (but important) to explain
Aguiar-Bergeron-Sottile saysA combinatorial Hopf algebra is a graded connected Hopf algebra with a multiplicative linear function.
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
partitions
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon
CHA’s in the mid-90s
partitions
CHA’s in the mid-90s
compositions
partitions
CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositions
CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositions
Commutative algebraof Quasi-symmetric functions
CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositionsGessel (ʻ84)
Commutative algebraof Quasi-symmetric functions
CHA’s in the mid-90s
FQSym
CHA’s in the mid-90s
FQSym
CHA’s in the mid-90s
permutations
Malvenuto-Reutenauer (ʻ95)
FQSym
CHA’s in the 90s+
CHA’s in the 90s+
uniform block permutations
Aguiar-Orellana ‘05
CHA’s in the 90s+
uniform block permutations
Aguiar-Orellana ‘05
14
1 2 3
4 5
6
7
8
9
10
11
12
13
Poirier-Reutenauer ‘95
Tableaux
CHA’s in the 90s+binary trees Connes-Kreimer ’98
Grossman-Larson ‘89
Loday-Ronco ‘98
uniform block permutations
Aguiar-Orellana ‘05
14
1 2 3
4 5
6
7
8
9
10
11
12
13
Poirier-Reutenauer ‘95
Tableaux
CHA’s in the 90s+CHA’s in the 90s+
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
Packed Words
Hivert ‘99
Set Compositions
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
Packed Words
Hivert ‘99
Set Compositions
1
13
2
3
23
Parking Functions
Novelli-Thibon ‘04
A zoo of Hopf Algberas
14
1 2 3
4 5
6
7
8
9
10
11
12
13
Hopf’sgraded
1
1
3
2
3
2
3
A goal of research on graded Hopf algebras?
One Goal of determining the Hopf algebras associated to combinatorial objects is to try and arrive at a classification theorem for Graded combinatorial Hopf algebras
14
1 2 3
4 5
6
7
8
9
10
11
12
13
A CLassification Theorem
A goal of research on graded Hopf algebras?
One Goal of determining the Hopf algebras associated to combinatorial objects is to try and arrive at a classification theorem for Graded combinatorial Hopf algebras
14
1 2 3
4 5
6
7
8
9
10
11
12
13
A CLassification Theorem
Aguiar Species CHAs
A goal of research on graded Hopf algebras?
One Goal of determining the Hopf algebras associated to combinatorial objects is to try and arrive at a classification theorem for Graded combinatorial Hopf algebras
14
1 2 3
4 5
6
7
8
9
10
11
12
13
A CLassification Theorem
Aguiar Species CHAs
(N)Bergeron-Lam-LiRep theory
Hopf algebras of setpartitions/compositions
Hopf algebras of setpartitions/compositions
Hopf algebras of setpartitions/compositions
Set composition definition:
Hopf algebras of setpartitions/compositions
Set composition definition:
Set partition definition:
Another type of non-commutative symmetric functions
Invariants under the left actionon the polynomial ring
Free algebra generated by one element at each degree
Wolf 1936, Rosas-Sagan ’03
Invariants under the left actionon the non-comm poly ring
Monomial set partition
For each monomial
Associate a set partition
Example:
Example:
Example:
Example:
This is a non commutative polynomial for a given
Example:
Consider the elements to be the object by letting the
number of variables in
This is a non commutative polynomial for a given
Combinatorial rule for product
Combinatorial rule for product
Combinatorial rule for product
CoproductInspiration:
CoproductInspiration:
CoproductInspiration:
Definition of
Non-commutativeCo-commutative
Hopf algebra of set partitions
Analogy between Sym and NCSym
Properties of NCSym
Properties of NCSym
Non-commutative and co-commutative
Properties of NCSym
Non-commutative and co-commutative
Has bases analogous to power, elementary, homogeneous, monomial symmetric functions in the algebra of Sym (Rosas-Sagan ’03, Bergeron-Brlek)
Properties of NCSym
Non-commutative and co-commutative
Has bases analogous to power, elementary, homogeneous, monomial symmetric functions in the algebra of Sym (Rosas-Sagan ’03, Bergeron-Brlek)
The dimension of the part of degree n are the bell numbers
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
Some questions to ask
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
What is the connection with representation theory?
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
What is the connection with representation theory?
What is the structure of this algebra?
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
What is the connection with representation theory?
What is the structure of this algebra?
What is the relationship with the ‘other’ non-commutative symmetric functions? (NSym)
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